An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation_7 pdf

First is the moneyness ratio m := log Se r T −t To interpret m, we need to generalize 6.11 into the formula Se µT −tfor the ex-pected value of the asset at expiry, given asset price S a

Fig 11.4 European put: Black–Scholes surface with asset path superimposed.

E

0

T

S t

delta

Fig 11.5 Black–Scholes surface for delta with three asset paths superimposed.

110 More on the Black–Scholes formulas

We will introduce three new dimensionless quantities First is the moneyness ratio

m := log Se r (T −t)

To interpret m, we need to generalize (6.11) into the formula Se µ(T −t)for the

ex-pected value of the asset at expiry, given asset price S at time t, Now we make the

assumption that the asset growth rate equals the interest rate,µ = r This

assump-tion will be examined in detail in Chapter 12; for now, we simply note that it leads

to the following conclusions

If m > 0, then the expected asset value at expiry is greater than the strike price In a

‘risk-neutral expectation at expiry’ sense, a call option is in-the-money and a put option

is out-of-the-money.

If m= 0, then, in the same sense, call and put options are at-the-money.

If m < 0, then, in the same sense, a call option is out-of-the-money and a put option is

in-the-money.

Second, we have the scaled volatility

τ := σ√T − t.

Here, the volatility is combined with the square root of the time to expiry This

is natural, since, for example, volatility appears in the form σ2(t i+1−t i ) in the

underlying asset model (6.9) The third step is to scale the option values by theasset price, by letting

c := C

S , for a call option,

and

p := P

S , for a put option.

In these new variables, d1and d2in (8.20) and (8.21) simplify to

11.7 Program of Chapter 11 and walkthrough 111

11.6 Notes and references

Colour versions of Figures 11.3, 11.4 and 11.5 can be downloaded from this book’swebsite, mentioned in the preface

aver-greater than max(S(t) − E, 0).

Is this argument valid?

11.2.  Show how Exercise 10.7 provides a counterexample to the following

statement:

As t goes from 0 to T , the Black–Scholes European put option value always

ap-proaches the payoff hockey-stick function from below.

11.3.  Verify (11.1) and (11.2).

11.4.    In the case where the volatility, σ, is zero in the asset model (6.9), the final asset price is the nonrandom quantity S0e µT The payoff from a Eu-

ropean option is then guaranteed to be max(S0e µT − E, 0) It may thus be

argued that the time-zero option value must be e −rTmax(S0e µT − E, 0).

However, this value clearly depends uponµ, whilst the Black–Scholes

for-mula does not (In fact, looking ahead to (14.2), the Black–Scholes value is

e −rTmax(S0e r T − E, 0).) Can you resolve this apparent contradiction?

11.5.  Show that ‘Call(−σ) = −Put(σ)’, that is, replacing σ in (8.19) by −σ

is equivalent to evaluating−P(S, t) in (8.24) This relation is sometimes called put–call supersymmetry.

11.7 Program of Chapter 11 and walkthrough

The program ch11 plots the Black–Scholes surface above the(S, t)-plane for a European call, in

the style of Figure 11.3 It is listed in Figure 11.6 We initialize E,r,sigma and T, and set up the array Svals of 50 equally spaced asset prices between 0 and 3 and the array tvals of 50 equally

spaced time points between 0 and T The nested for loops then work through Svals and tvals,

using ch08 to evaluate the Black–Scholes formula The European call value is stored in the dimensional array Call We then use meshgrid to set up two-dimensional arrays Smat and tmat that are appropriate for use with the three-dimensional plotting function mesh.

two-112 More on the Black–Scholes formulas

%CH11 Program for Chapter 11

ylabel(’S’), xlabel(’t’), zlabel(’C(S,t)’)

Fig 11.6 Program of Chapter 11: ch11.m.

P R O G R A M M I N G E X E R C I S E S

P11.1 Editch11.m so that it applies to a European put option, as in Figure 11.4

P11.2 Edit ch11.m so that it applies to the delta of a European call option, as

in Figure 11.5, and investigate the use ofsurf, surfc and waterfall instead ofmesh

Quotes

The Black–Scholes formula is still around,

even though it depends on at least 10 unrealistic assumptions.

Making the assumptions more realistic

hasn’t produced a formula that works better across a wide range of circumstances.

F I S C H E R B L A C K (Black, 1989)

We know this doesn’t work by rote.

But this is the best model we have.

You look at the old-timers who went with their gut.

You had this model, you had these numbers,

11.7 Program of Chapter 11 and walkthrough 113 and in the end you thought they were a lot more powerful than a guy’s gut.

ROBERT STAVIS , former member of the Arbitrage group at Salomon Brothers, source

(Lowenstein, 2001)

A first-rate theory predicts,

a second-rate theory forbids

and a third-rate theory explains after the event.

A L E X A N D E R K I T A I G O R O D S K I , 1975, source www.byrneweb.com/sunburn/quotes html

In the days before the Black–Scholes formula, it was often argued that a reasonable

way to value an option is to take the expected payoff In this chapter we show how

the expected payoff idea fits in with the Black–Scholes methodology This leads us

to the concept of risk neutrality, which will play a fundamental role in Chapters 15,

16 and beyond, when we discuss computational algorithms

12.2 Expected payoff

To cover European call and put options in a single notation, we let
(x) denote the

payoff function, so
(x) = max(x − E, 0) for a call and
(x) = max(E − x, 0) for a put The treatment here easily generalizes to other European-style options,

that is, options whose payoff may be expressed as a function of the asset price atexpiry

Under our model (6.8), the final asset price, S (T ), is a random variable of the form S (T ) = S0e (µ−σ2/2)T +σ√T Z , where Z ∼ N(0, 1) So the payoff,
(S(T )),

is also a known random variable Why don’t we simply take the time-zero optionvalue to be the average payoff, suitably discounted for interest? This gives a value

On the face of it, expected payoff seems to have no place in option valuationtheory However, by a remarkable twist, it is possible to rehabilitate the idea.

12.3 Risk neutrality

Figure 12.1 confirms that the time-zero discounted expected payoff (12.2) is indeed

a function ofµ The solid line plots (12.2) as µ varies from 0 to 0.1 for a European call with S0= 10, E = 9, r = 0.05, σ = 0.2 and T = 3 As we would guess, the

expected payoff increases with the growth rate,µ Superimposed on the picture as

a dashed line is the Black–Scholes option value, 2.66.

12.3 Risk neutrality 117

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 1.5

coinci-W (S, t) in (12.4) satisfies the Black–Scholes PDE (8.15) when µ = r.

Now we check the final time and boundary conditions Taking t = T in (12.3),

we note that if S (T ) is given, and thus nonrandom, then E(
(S(T ))) =
(S(T )),

giving

W (S, T ) =
(S(T )).

Hence the conditions (8.16) for a call and (8.25) for a put are satisfied Similarly,

if S = 0 at any time then we know from (6.9) that S(T ) = 0, and hence in (12.3)

W (0, t) = e −r(T −t)
(0).

This matches (8.17) and (8.26) for the call and put, respectively Finally, we notethat the arguments given to justify (8.18) and (8.27) are equally valid for (12.3)

Overall, since W (S, t) with µ = r satisfies the same PDE and the same final

time/ boundary conditions, the uniqueness of the solution tells us that

118 Risk neutrality

W (S, t) in (12.4) reproduces the Black–Scholes option value when µ = r.

We could re-write this conclusion as follows

No matter what parametersµ and σ in the asset model (6.9) we believe to be correct, we

can obtain the Black–Scholes option value by pretending that the drift,µ, is equal to the interest rate, r , and taking the discounted expected payoff.

In settingµ = r we are making what is known as a risk neutrality assumption.

We will see in Chapters 15 and 16 that the risk-neutral expectation frameworkallows us to develop computational methods for approximating options where an-alytical formulas are not available

12.4 Notes and references

It is perfectly standard, but not particularly enlightening, to give the name risk neutrality to the condition µ = r The phrase borrows from the concept of a risk- neutral investor; an unlikely person who regards

• an investment with guaranteed rate of return r, and

• a risky investment with expected rate of return r

as equally attractive In the case where all assets satisfy the lognormal model (6.9)with the same growth parameterµ – the so-called risk-neutral world – we see from

(6.11) that a risk-neutral investor would have no preferences between investing in

a bank and in any asset

In the risk-neutral world, (6.11) shows that E(S(t)) = S0e r t, so the expecteddiscounted asset price is E(e −rt S (t)) = S0 In other words, the expected dis-counted asset price does not change with time; it remains at its time-zero level Aprocess like this, whose expected future value is given by its current value, is called

a martingale By using martingale theory it is possible to convert the simple

obser-vation in Exercise 12.1 into a rigorous and powerful theory for option valuation

In particular, this is an alternative way to derive the Black–Scholes formulas Thetexts (Duffie, 2001; Karatzas and Shreve, 1998; Nielsen, 1999) cover this material

in depth, while perhaps the most accessible introduction is (Baxter and Rennie,1996) Chapter 6 of (Kritzman, 2000) also gives a very readable, example-drivencoverage of risk neutrality

In Chapter 16 we introduce the binomial method as a computational techniquefor option valuation It is also possible to use the binomial framework as ananalytical tool with which the Black–Scholes formulas can be derived withoutrecourse to PDEs The concept of risk neutrality arises quite naturally in thissetting Exercise 12.5 provides a cut-down version of the idea The text (Baxter

12.4 Notes and references 119and Rennie, 1996) and the on-line lecture notes of Professor Robert Kohn atwww.math.nyu.edu/faculty/kohn/ are good places to learn more.

E X E R C I S E S

12.1.    Using a large sheet of paper and a pen with plenty of ink, show that

for µ = r the quantity W(S, t) in (12.4) satisfies the Black–Scholes PDE

(8.15) (You may differentiate inside the integral sign without worryingabout whether this is justified.)

12.2.    Consider a European-style option with payoff at expiry given by

(S(T )) = S(T ) Explain why the time-zero value of this option must be

S0 By using (6.11), show that asking for the discounted expected payoff(12.1) to match this value leads immediately to the risk neutrality condition

µ = r.

12.3.  Given initial asset price S0 at time t = 0, show that, in a risk-neutral

world, the factor N (d2) in the Black–Scholes formula (8.19) represents the

probability that a European call option will be exercised

12.4.    Show that the value W(S, t) in (12.4) can be computed from the

fol-lowing recipe

(i) Compute the Black–Scholes option value at(S, t) with the interest rate set

to r = µ.

(ii) Scale this quantity by e (µ−r)(T −t).

(This recipe was used to create Figure 12.1.)

12.5.    Consider the following, simplified scenario for valuing a

European-style option

• The time-zero asset price is S0

• At expiry, the asset price may take only two possible values

S (T ) = Sup > S0, with probability p ,

S (T ) = Sdown< S0, with probability 1− p.

Let
denote the payoff function, and let
up :=
(Sup) and
down :=

(Sdown) denote the two possible payoffs at expiry Take a portfolio at time

t = 0 consisting of A units of asset and an amount C of cash Asking for this portfolio to replicate the option (i.e to have payoff
upwhen S (T ) =

Sup and
down when S (T ) = Sdown) leads to a pair of linear equations for

A and C Find and solve these to obtain

A=
up−
down

Use the no arbitrage principle to argue that 0< q < 1 must hold Show

that the value in (12.7) may also be interpreted as the discounted expectedpayoff of an asset taking the values

S (T ) = Sup > S0, with probability q ,

S (T ) = Sdown < S0, with probability 1− q.

Can you see any features from this simplified scenario that carry through

to the Black–Scholes version?

12.6.    In Section 10.3 we gave a financial interpretation of the inequality ρ >

0 Use the risk neutrality viewpoint to give an alternative interpretation

12.5 Program of Chapter 12 and walkthrough

The program ch12, listed in Figure 12.2, illustrates risk neutrality in the manner of Figure 12.1 We fix S,E,r,sigma and T and an array of 200 values for mu A for loop is then used to compute an array epayoff which stores the discounted time-zero Black–Scholes value when r is set to each mu value; see Exercise 12.4 This is done via the ch08 function from Chapter 8 After executing this loop, we use ch08 to obtain the true Black–Scholes value, C We then plot the (muvals,epayoff) curve and superimpose a dashed line at height C.

P R O G R A M M I N G E X E R C I S E S

P12.1 Confirm experimentally the result mentioned in Exercise 12.3 Do this by

generating a large number of expiry-time asset prices, and counting the proportionthat are in-the-money

P12.2 Investigate the use ofquad and quadl for evaluating integrals of the form(12.4)

12.5 Program of Chapter 12 and walkthrough 121

%CH12 Program for Chapter 12

%

% Compute expected payoff for European call

% Illustrates risk neutrality

% work out time-zero Black-Scholes value with r = mu

[C, Cdelta, P, Pdelta] = ch08(S,E,mu,sigma,T);

epayoff(k) = exp((mu-r)*T)*C;

end

% true Black–Scholes value

[C, Cdelta, P, Pdelta] = ch08(S,E,r,sigma,T);

plot(muvals,epayoff,’r-’);

hold on, grid on

plot([muvals(1),muvals(end)],[C,C],’b-’);

xlabel(’\mu’), legend(’Expected payoff’,’Black-Scholes’)

Fig 12.2 Program of Chapter 12: ch12.m.

Quotes

risk-neutrality is far from easy to grasp intuitively,

which is perhaps the source of the confusion above.

The key steps in the derivation of the Black–Scholes equation,

namely no arbitrage and that risk-free portfolios can earn the risk-free rate,

are intuitively clear.

P A U L W I L M O T T , S A M H O W I S O N A N D J E F F D E W Y N N E(Wilmott et al., 1995)

Risk neutral valuation, which was developed by John Cox and Stephen Ross,

has the dual virtues that it can be applied to practically any option valuation problem and it is marvelously intuitive.

M A R K P K R I T Z M A N (Kritzman, 2000)

To put it simply,

if there is an arbitrage price, any other price is too dangerous to quote.

M A R T I N B A X T E R A N D A N D R E W R E N N I E (Baxter and Rennie, 1996)

13.2 General problem

The task that we consider in this chapter is

given a function F : R → R, find an x
∈ R such that F(x
) = 0.

In general, of course, we cannot find an x
analytically, and must therefore

con-tent ourselves with an approximation via a computational method It is also worth

keeping in mind that, depending on the nature of F, there may be no suitable x
,

exactly one x
or many x
values.

13.3 Bisection

The bisection method is based on the observation that if a continuous function

changes sign then it must pass through zero; that is,

for continuous F, if x a < xb with F (xa)F(xb) < 0, then F (x
) = 0 for some xa < x
< xb.

Having found x a and x b with F (x a )F(x b ) < 0, we could evaluate F at the point xmid := (x a + x b )/2 The sign of F(xmid) must then match either F(x a ) or

mid-123